Expand all Collapse all | Results 101 - 125 of 226 |
101. CJM 2008 (vol 60 pp. 734)
Genus 2 Curves with Quaternionic Multiplication We explicitly construct the canonical rational models of Shimura
curves, both analytically in terms of modular forms and
algebraically in terms of coefficients of genus 2 curves, in the
cases of quaternion algebras of discriminant 6 and 10. This emulates
the classical construction in the elliptic curve case. We also give
families of genus 2 QM curves, whose Jacobians are the corresponding
abelian surfaces on the Shimura curve, and with coefficients that
are modular forms of weight 12. We apply these results to show
that our $j$-functions are supported exactly at those primes where
the genus 2 curve does not admit potentially good reduction, and
construct fields where this potentially good reduction is attained.
Finally, using $j$, we construct the fields of moduli and definition
for some moduli problems associated to the Atkin--Lehner group
actions.
Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli Categories:11G18, 14G35 |
102. CJM 2008 (vol 60 pp. 481)
Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields Let $k$ be a global field, $\overline{k}$ a separable
closure of $k$, and $G_k$ the absolute Galois group
$\Gal(\overline{k}/k)$ of $\overline{k}$ over $k$. For every
$\sigma\in G_k$, let $\ks$ be the fixed subfield of $\overline{k}$
under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known
that the Mordell--Weil group $E(\ks)$ has infinite rank. We present a
new proof of this fact in the following two cases. First, when $k$
is a global function field of odd characteristic and $E$ is
parametrized by a Drinfeld modular curve, and secondly when $k$ is a
totally real number field and $E/k$ is parametrized by a Shimura
curve. In both cases our approach uses the non-triviality of a
sequence of Heegner points on $E$ defined over ring class fields.
Category:11G05 |
103. CJM 2008 (vol 60 pp. 491)
A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations We solve several multi-parameter families of binomial Thue equations of arbitrary
degree; for example, we solve the equation
\[
5^u x^n-2^r 3^s y^n= \pm 1,
\]
in non-zero integers $x$, $y$ and positive integers $u$, $r$, $s$ and $n \geq 3$.
Our approach uses several Frey curves simultaneously, Galois representations
and level-lowering, new lower bounds for linear
forms in $3$ logarithms due to Mignotte and a famous theorem of Bennett on binomial
Thue equations.
Keywords:Diophantine equations, Frey curves, level-lowering, linear forms in logarithms, Thue equation Categories:11F80, 11D61, 11D59, 11J86, 11Y50 |
104. CJM 2008 (vol 60 pp. 532)
Local Bounds for Torsion Points on Abelian Varieties We say that an abelian variety over a $p$-adic field $K$ has
anisotropic reduction (AR) if the special fiber of its N\'eron minimal
model does not contain a nontrivial split torus. This includes all
abelian varieties with potentially good reduction and, in particular,
those with complex or quaternionic multiplication. We give a bound for
the size of the $K$-rational torsion subgroup of a $g$-dimensional AR
variety depending only on $g$ and the numerical invariants of $K$ (the
absolute ramification index and the cardinality of the residue
field). Applying these bounds to abelian varieties over a number field
with everywhere locally anisotropic reduction, we get bounds which, as
a function of $g$, are close to optimal. In particular, we determine
the possible cardinalities of the torsion subgroup of an AR abelian
surface over the rational numbers, up to a set of 11 values which are
not known to occur. The largest such value is 72.
Categories:11G10, 14K15 |
105. CJM 2008 (vol 60 pp. 412)
Quelques calculs de traces compactes et leurs transform{Ã©es de Satake On calcule les restrictions {\`a} l'alg{\`e}bre de Hecke sph{\'e}rique
des traces tordues compactes d'un ensemble de repr{\'e}sentations
explicitement construites du groupe $\GL(N, F)$, o{\`u} $F$ est
un corps $p$-adique. Ces calculs r\'esolve en particulier une
question pos{\'e}e dans un article pr\'ec\'edent du m\^eme auteur.
Categories:22E50, 11F70 |
106. CJM 2008 (vol 60 pp. 208)
Constructing Galois Representations with Very Large Image Starting with a 2-dimensional mod $p$ Galois representation, we
construct a deformation to a power series ring in infinitely many
variables over the $p$-adics. The image of this representation is full
in the sense that it contains $\SL_2$ of this power series
ring. Furthermore, all ${\mathbb Z}_p$ specializations of this
deformation are potentially semistable at $p$.
Keywords:Galois representation, deformation Category:11f80 |
107. CJM 2007 (vol 59 pp. 1323)
On a Conjecture of Jacquet, Lai, and Rallis: Some Exceptional Cases We prove two spectral identities. The first one relates the relative
trace formula for the spherical variety $\GSpin(4,3)/G_2$ with a
weighted trace formula for $\GL_2$. The second relates a spherical
variety pertaining to $F_4$ to one of $\GSp(6)$. These identities are
in accordance with a conjecture made by Jacquet, Lai, and Rallis,
and are obtained without an appeal to a geometric comparison.
Categories:11F70, 11F72, 11F30, 11F67 |
108. CJM 2007 (vol 59 pp. 1121)
Meromorphic Continuation of Spherical Cuspidal Data Eisenstein Series Meromorphic continuation of the Eisenstein series induced from spherical,
cuspidal data on parabolic subgroups is achieved via reworking
Bernstein's adaptation of Selberg's third proof of meromorphic
continuation.
Categories:11F72, 32N10, 32D15 |
109. CJM 2007 (vol 59 pp. 1284)
On Effective Witt Decomposition and the Cartan--Dieudonn{Ã© Theorem Let $K$ be a number field, and let $F$ be a symmetric bilinear form in
$2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical
theorem of Witt states that the bilinear space $(Z,F)$ can be
decomposed into an orthogonal sum of hyperbolic planes and singular and
anisotropic components. We prove the existence of such a decomposition
of small height, where all bounds on height are explicit in terms of
heights of $F$ and $Z$. We also prove a special version of Siegel's
lemma for a bilinear space, which provides a small-height orthogonal
decomposition into one-dimensional subspaces. Finally, we prove an
effective version of the Cartan--Dieudonn{\'e} theorem. Namely, we show
that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can
be represented as a product of reflections of bounded heights with an
explicit bound on heights in terms of heights of $F$, $Z$, and
$\sigma$.
Keywords:quadratic form, heights Categories:11E12, 15A63, 11G50 |
110. CJM 2007 (vol 59 pp. 1050)
On the Restriction to $\D^* \times \D^*$ of Representations of $p$-Adic $\GL_2(\D)$ Let $\mathcal{D}$ be a division algebra
over a nonarchimedean local field. Given
an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we
describe its restriction to the diagonal subgroup $\mathcal{D}^* \times
\mathcal{D}^*$. The description is in terms of the structure of the
twisted Jacquet module of the representation $\pi$. The proof
involves Kirillov theory that we have developed earlier in joint work
with Dipendra Prasad. The main result on restriction also shows that
$\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$-distinguished if and only if
$\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$
is a quaternion division algebra then the twisted Jacquet module
is multiplicity-free by proving an appropriate theorem on invariant
distributions; this then proves a multiplicity-one theorem on the
restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic
case.
Categories:22E50, 22E35, 11S37 |
111. CJM 2007 (vol 59 pp. 673)
Hecke $L$-Functions and the Distribution of Totally Positive Integers Let $K$ be a totally real number field of degree $n$. We show that
the number of totally positive integers
(or more generally the number of totally positive elements of a given fractional ideal)
of given trace is evenly distributed around its expected value, which is
obtained from geometric considerations.
This result depends on unfolding an integral over
a compact torus.
Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace Categories:11M41, 11F30, , 11F55, 11H06, 11R47 |
112. CJM 2007 (vol 59 pp. 503)
Cyclic Groups and the Three Distance Theorem We give a two dimensional extension of the three distance Theorem. Let
$\theta$ be in $\mathbf{R}^{2}$ and let $q$ be in $\mathbf{N}$. There exists a
triangulation of $\mathbf{R}^{2}$ invariant by $\mathbf{Z}^{2}$-translations,
whose set of vertices is $\mathbf{Z}^{2}+\{0,\theta,\dots,q\theta\}$, and whose
number of different triangles, up to translations, is bounded above by a
constant which does not depend on $\theta$ and $q$.
Categories:11J70, 11J71, 11J13 |
113. CJM 2007 (vol 59 pp. 553)
Computations of Elliptic Units for Real Quadratic Fields Let $K$ be a real quadratic field, and $p$ a rational prime which is
inert in $K$. Let $\alpha$ be a modular unit on $\Gamma_0(N)$. In an
earlier joint article with Henri Darmon, we presented the definition
of an element $u(\alpha, \tau) \in K_p^\times$ attached to $\alpha$
and each $\tau \in K$. We conjectured that the $p$-adic number
$u(\alpha, \tau)$ lies in a specific ring class extension of $K$
depending on $\tau$, and proposed a ``Shimura reciprocity law"
describing the permutation action of Galois on the set of $u(\alpha,
\tau)$. This article provides computational evidence for these
conjectures. We present an efficient algorithm for computing
$u(\alpha, \tau)$, and implement this algorithm with the modular unit
$\alpha(z) = \Delta(z)^2\Delta(4z)/\Delta(2z)^3.$ Using $p = 3, 5, 7,$
and $11$, and all real quadratic fields $K$ with discriminant $D <
500$ such that $2$ splits in $K$ and $K$ contains no unit of negative
norm, we obtain results supporting our conjectures. One of the
theoretical results in this paper is that a certain measure used to
define $u(\alpha, \tau)$ is shown to be $\mathbf{Z}$-valued rather
than only $\mathbf{Z}_p \cap \mathbf{Q}$-valued; this is an
improvement over our previous result and allows for a precise
definition of $u(\alpha, \tau)$, instead of only up to a root of
unity.
Categories:11R37, 11R11, 11Y40 |
114. CJM 2007 (vol 59 pp. 372)
Zeta Functions of Supersingular Curves of Genus 2 We determine which isogeny classes of supersingular abelian
surfaces over a finite field $k$ of characteristic $2$ contain
jacobians. We deal with this problem in a direct way by computing
explicitly the zeta function of all supersingular curves of genus
$2$. Our procedure is constructive, so that we are able to exhibit
curves with prescribed zeta function and find formulas for the
number of curves, up to $k$-isomorphism, leading to the same zeta
function.
Categories:11G20, 14G15, 11G10 |
115. CJM 2007 (vol 59 pp. 127)
Smooth Values of the Iterates of the Euler Phi-Function Let $\phi(n)$ be the Euler phi-function, define
$\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all
$k\geq 0$. We will determine an asymptotic formula for the set of
integers $n$ less than $x$ for which $\phi_k(n)$ is $y$-smooth,
conditionally on a weak form of the Elliott--Halberstam conjecture.
Categories:11N37, 11B37, 34K05, 45J05 |
116. CJM 2007 (vol 59 pp. 211)
On Two Exponents of Approximation Related to a Real Number and Its Square For each real number $\xi$, let $\lambdahat_2(\xi)$ denote the
supremum of all real numbers $\lambda$ such that, for each
sufficiently large $X$, the inequalities $|x_0| \le X$,
$|x_0\xi-x_1| \le X^{-\lambda}$ and $|x_0\xi^2-x_2| \le
X^{-\lambda}$ admit a solution in integers $x_0$, $x_1$ and $x_2$
not all zero, and let $\omegahat_2(\xi)$ denote the supremum of
all real numbers $\omega$ such that, for each sufficiently large
$X$, the dual inequalities $|x_0+x_1\xi+x_2\xi^2| \le
X^{-\omega}$, $|x_1| \le X$ and $|x_2| \le X$ admit a solution in
integers $x_0$, $x_1$ and $x_2$ not all zero. Answering a
question of Y.~Bugeaud and M.~Laurent, we show that the exponents
$\lambdahat_2(\xi)$ where $\xi$ ranges through all real numbers
with $[\bQ(\xi)\wcol\bQ]>2$ form a dense subset of the interval $[1/2,
(\sqrt{5}-1)/2]$ while, for the same values of $\xi$, the dual
exponents $\omegahat_2(\xi)$ form a dense subset of $[2,
(\sqrt{5}+3)/2]$. Part of the proof rests on a result of
V.~Jarn\'{\i}k showing that $\lambdahat_2(\xi) =
1-\omegahat_2(\xi)^{-1}$ for any real number $\xi$ with
$[\bQ(\xi)\wcol\bQ]>2$.
Categories:11J13, 11J82 |
117. CJM 2007 (vol 59 pp. 148)
On Certain Classes of Unitary Representations for Split Classical Groups In this paper we prove the unitarity of duals of tempered
representations supported on minimal parabolic subgroups for split
classical $p$-adic groups. We also construct a family of unitary
spherical representations for real and complex classical groups
Categories:22E35, 22E50, 11F70 |
118. CJM 2007 (vol 59 pp. 85)
On the Convergence of a Class of Nearly Alternating Series Let $C$ be the class of convex sequences of real numbers. The
quadratic irrational numbers can be partitioned into two types as
follows. If $\alpha$ is of the first type and $(c_k) \in C$, then
$\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if
$c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and
$(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$
converges if and only if $\sum c_k/k$ converges. An example of a
quadratic irrational of the first type is $\sqrt{2}$, and an
example of the second type is $\sqrt{3}$. The analysis of this
problem relies heavily on the representation of $ \alpha$ as a
simple continued fraction and on properties of the sequences of
partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$
and double partial sums $T(n)=\sum_{k=1}^n S(k)$.
Keywords:Series, convergence, almost alternating, convex, continued fractions Categories:40A05, 11A55, 11B83 |
119. CJM 2006 (vol 58 pp. 1203)
Orbites unipotentes et pÃ´les d'ordre maximal de la fonction $\mu $ de Harish-Chandra Dans un travail ant\'erieur, nous
avions montr\'e que l'induite parabolique (normalis\'ee) d'une
repr\'esentation irr\'eductible cuspidale $\sigma $ d'un
sous-groupe de Levi $M$ d'un groupe $p$-adique contient un
sous-quotient de carr\'e int\'egrable, si et seulement si la
fonction $\mu $ de Harish-Chandra a un p\^ole en $\sigma $ d'ordre
\'egal au rang parabolique de $M$. L'objet de cet article est
d'interpr\'eter ce r\'esultat en termes de fonctorialit\'e de
Langlands.
Categories:11F70, 11F80, 22E50 |
120. CJM 2006 (vol 58 pp. 1095)
A Casselman--Shalika Formula for the Shalika Model of $\operatorname{GL}_n$ The Casselman--Shalika method is a way to compute explicit
formulas for periods of irreducible unramified representations of
$p$-adic groups that are associated to unique models (i.e.,
multiplicity-free induced representations). We apply this method
to the case of the Shalika model of $GL_n$, which is known to
distinguish lifts from odd orthogonal groups. In the course of our
proof, we further develop a variant of the method, that was
introduced by Y. Hironaka, and in effect reduce many such problems
to straightforward calculations on the group.
Keywords:Casselman--Shalika, periods, Shalika model, spherical functions, Gelfand pairs Categories:22E50, 11F70, 11F85 |
121. CJM 2006 (vol 58 pp. 796)
Mordell--Weil Groups and the Rank of Elliptic Curves over Large Fields Let $K$ be a number field, $\overline{K}$ an algebraic closure of
$K$ and $E/K$ an elliptic curve
defined over $K$. In this paper, we prove that if $E/K$ has a
$K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then
for each $\sigma\in \Gal(\overline{K}/K)$, the Mordell--Weil group
$E(\overline{K}^{\sigma})$ of $E$ over the fixed subfield of
$\overline{K}$ under $\sigma$ has infinite rank.
Category:11G05 |
122. CJM 2006 (vol 58 pp. 843)
On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions In a previous article, we studied the distribution of ``low-lying"
zeros of the family of quad\-ratic Dirichlet $L$-functions assuming
the Generalized Riemann Hypothesis for all Dirichlet
$L$-functions. Even with this very strong assumption, we were
limited to using weight functions whose Fourier transforms are
supported in the interval $(-2,2)$. However, it is widely believed
that this restriction may be removed, and this leads to what has
become known as the One-Level Density Conjecture for the zeros of
this family of quadratic $L$-functions. In this note, we make use
of Weil's explicit formula as modified by Besenfelder to prove an
analogue of this conjecture.
Category:11M26 |
123. CJM 2006 (vol 58 pp. 643)
Centralizers and Twisted Centralizers: Application to Intertwining Operators ABSTRACT
The equality of the centralizer and twisted centralizer is proved
based on a case-by-case analysis when the unipotent radical of a
maximal parabolic subgroup is abelian.
Then this result is used to determine the poles of intertwining operators.
Category:11F70 |
124. CJM 2006 (vol 58 pp. 580)
Annihilators for the Class Group of a Cyclic Field of Prime Power Degree, II We prove, for a field $K$ which is cyclic of odd prime power
degree over the rationals, that the annihilator of the
quotient of the units of $K$ by a suitable large subgroup (constructed
from circular units) annihilates what we call the
non-genus part of the class group.
This leads to stronger annihilation results for the whole
class group than a routine application of the Rubin--Thaine method
would produce, since the
part of the class group determined by genus theory has an obvious
large annihilator which is not detected by
that method; this is our reason for concentrating on
the non-genus part. The present work builds on and strengthens
previous work of the authors; the proofs are more conceptual now,
and we are also able to construct an example which demonstrates
that our results cannot be easily sharpened further.
Categories:11R33, 11R20, 11Y40 |
125. CJM 2006 (vol 58 pp. 419)
Stark's Conjecture and New Stickelberger Phenomena We introduce a new conjecture concerning the construction
of elements in the annihilator ideal
associated to a Galois action on the higher-dimensional algebraic
$K$-groups of rings of integers in number fields. Our conjecture is
motivic in the sense that it involves the (transcendental) Borel
regulator as well as being related to $l$-adic \'{e}tale
cohomology. In addition, the conjecture generalises the well-known
Coates--Sinnott conjecture. For example, for a totally real
extension when $r = -2, -4, -6, \dotsc$ the Coates--Sinnott
conjecture merely predicts that zero annihilates $K_{-2r}$ of the
ring of $S$-integers while our conjecture predicts a non-trivial
annihilator. By way of supporting evidence, we prove the
corresponding (conjecturally equivalent) conjecture for the Galois
action on the \'{e}tale cohomology of the cyclotomic extensions of
the rationals.
Categories:11G55, 11R34, 11R42, 19F27 |